From archgoon’s link:
> "This puzzle confounds people legitimately, however, because most of the ways in which you are likely to find out that X has at least one boy contain an implicit bias which changes the answer.”
Exactly right. The phrasing in the problem asks the reader to consider a particular scenario, not isolated facts. Consider what happens if you take the question at its word:
> If a person on the street says to you, "I have two children," what is the probability that they're both boys?
Hell if I know. Maybe one is a boy and the other is a grown man? Or maybe he has five children, BBGGG. “I have two children, both of which are boys” is technically correct if there’s two boys and three girls, right? Or maybe he’s just outright lying! How do I assign a probability to any of those?
Consider this phrasing instead:
“If you flip a coin twice, and the first and/or the second is heads, what’s the probability that both are heads?” I’d expect more people to get this right than the boy/girl question.
Also, as an aside, punctuation is important:
“I have two children. At least, one of them is a boy.”
(Yikes!)